LisMath Seminar 

26/04/2021, 11:30 — 12:00 — Online
Vicente Lorenzo García, LisMath, Instituto Superior Técnico, Universidade de Lisboa

Group actions on surfaces of general type and moduli spaces

The main numerical invariants of a complex projective algebraic surface $X$ are the self-intersection of its canonical class $K^2_X$ and its holomorphic Euler characteristic $\chi(\mathcal{O}_X)$. If we assume $X$ to be minimal and of general type then $K^2_X\geq 2\chi(\mathcal{O}_X)-6$ by Noether's inequality.

Minimal algebraic surfaces of general type $X$ such that $K^2_X=2\chi(\mathcal{O}_X)-6$ or $K^2_X=2\chi(\mathcal{O}_X)-5$ are called Horikawa surfaces and they admit a canonical $\mathbb{Z}_2$-action. In this talk we will discuss other possible group actions on Horikawa surfaces. In particular, $\mathbb{Z}_2^2$-actions and $\mathbb{Z}_3$-actions on Horikawa surfaces will be studied.

In the case of $\mathbb{Z}_2^2$-actions we will not settle for Horikawa surfaces and results regarding the geography of minimal surfaces of general type admitting a $\mathbb{Z}_2^2$-action will be discussed. They will yield some consequences on the moduli spaces of stable surfaces $\overline{\mathfrak{M}}_{K^2,\chi}$.